C2xHe3.4S3 - GroupNames

G = C₂×He₃.₄S₃ order 324 = 2²·3⁴

Direct product of C₂ and He₃.₄S₃

direct product, metabelian, supersoluble, monomial

Aliases: C₂×He₃.₄S₃, He₃.₇D₆, 3_-¹⁺²⋊₅D₆, C₉.(S₃×C₆), C₉⋊S₃⋊₆C₆, (C₃×C₁₈)⋊₇S₃, (C₃×C₁₈)⋊₆C₆, (C₃×C₉)⋊₁₁D₆, C₁₈.₉(C₃×S₃), C₃².₈(S₃×C₆), C₉○He₃⋊₂C₂², (C₂×He₃).₁₃S₃, (C₂×3_-¹⁺²)⋊₄S₃, (C₂×C₉⋊S₃)⋊₅C₃, (C₃×C₉)⋊₇(C₂×C₆), C₃.₄(C₆×C₃⋊S₃), C₆.₈(C₃×C₃⋊S₃), (C₃×C₆).₂₁(C₃×S₃), (C₂×C₉○He₃)⋊₁C₂, C₃².₆(C₂×C₃⋊S₃), (C₃×C₆).₁₄(C₃⋊S₃), SmallGroup(324,147)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₉ — C₂×He₃.₄S₃

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₉○He₃ — He₃.₄S₃ — C₂×He₃.₄S₃

Lower central

C₃×C₉ — C₂×He₃.₄S₃

Upper central

C₁ — C₂

Generators and relations for C₂×He₃.₄S₃
G = < a,b,c,d,e,f | a²=b³=c³=d³=f²=1, e³=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd^-1=bc^-1, be=eb, fbf=b^-1, cd=dc, ce=ec, fcf=c^-1, de=ed, df=fd, fef=c^-1e² >

Subgroups: 457 in 93 conjugacy classes, 32 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², S₃, C₆, C₆, C₉, C₉, C₉, C₃², C₃², D₆, C₂×C₆, D₉, C₁₈, C₁₈, C₁₈, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₉, C₃×C₉, He₃, 3_-¹⁺², 3_-¹⁺², D₁₈, S₃×C₆, C₂×C₃⋊S₃, C₃×D₉, C₃²⋊C₆, C₉⋊C₆, C₉⋊S₃, C₃×C₁₈, C₃×C₁₈, C₂×He₃, C₂×3_-¹⁺², C₂×3_-¹⁺², C₉○He₃, C₆×D₉, C₂×C₃²⋊C₆, C₂×C₉⋊C₆, C₂×C₉⋊S₃, He₃.₄S₃, C₂×C₉○He₃, C₂×He₃.₄S₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₆, C₂×C₆, C₃×S₃, C₃⋊S₃, S₃×C₆, C₂×C₃⋊S₃, C₃×C₃⋊S₃, C₆×C₃⋊S₃, He₃.₄S₃, C₂×He₃.₄S₃

Smallest permutation representation of C₂×He₃.₄S₃
►On 54 points

Generators in S₅₄

(1 52)(2 53)(3 54)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 35)(11 36)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 41)(20 42)(21 43)(22 44)(23 45)(24 37)(25 38)(26 39)(27 40)

(1 42 33)(2 43 34)(3 44 35)(4 45 36)(5 37 28)(6 38 29)(7 39 30)(8 40 31)(9 41 32)(10 54 22)(11 46 23)(12 47 24)(13 48 25)(14 49 26)(15 50 27)(16 51 19)(17 52 20)(18 53 21)

(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)

(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)

(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)

(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 54)(9 53)(10 40)(11 39)(12 38)(13 37)(14 45)(15 44)(16 43)(17 42)(18 41)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 36)(27 35)

G:=sub<Sym(54)| (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,35)(11,36)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,41)(20,42)(21,43)(22,44)(23,45)(24,37)(25,38)(26,39)(27,40), (1,42,33)(2,43,34)(3,44,35)(4,45,36)(5,37,28)(6,38,29)(7,39,30)(8,40,31)(9,41,32)(10,54,22)(11,46,23)(12,47,24)(13,48,25)(14,49,26)(15,50,27)(16,51,19)(17,52,20)(18,53,21), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,54)(9,53)(10,40)(11,39)(12,38)(13,37)(14,45)(15,44)(16,43)(17,42)(18,41)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,36)(27,35)>;

G:=Group( (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,35)(11,36)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,41)(20,42)(21,43)(22,44)(23,45)(24,37)(25,38)(26,39)(27,40), (1,42,33)(2,43,34)(3,44,35)(4,45,36)(5,37,28)(6,38,29)(7,39,30)(8,40,31)(9,41,32)(10,54,22)(11,46,23)(12,47,24)(13,48,25)(14,49,26)(15,50,27)(16,51,19)(17,52,20)(18,53,21), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,54)(9,53)(10,40)(11,39)(12,38)(13,37)(14,45)(15,44)(16,43)(17,42)(18,41)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,36)(27,35) );

G=PermutationGroup([[(1,52),(2,53),(3,54),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,35),(11,36),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,41),(20,42),(21,43),(22,44),(23,45),(24,37),(25,38),(26,39),(27,40)], [(1,42,33),(2,43,34),(3,44,35),(4,45,36),(5,37,28),(6,38,29),(7,39,30),(8,40,31),(9,41,32),(10,54,22),(11,46,23),(12,47,24),(13,48,25),(14,49,26),(15,50,27),(16,51,19),(17,52,20),(18,53,21)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33),(37,40,43),(38,41,44),(39,42,45)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,54),(9,53),(10,40),(11,39),(12,38),(13,37),(14,45),(15,44),(16,43),(17,42),(18,41),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,36),(27,35)]])

42 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	9A	9B	9C	9D	···	9K	18A	18B	18C	18D	···	18K
order	1	2	2	2	3	3	3	3	3	3	6	6	6	6	6	6	6	6	6	6	9	9	9	9	···	9	18	18	18	18	···	18
size	1	1	27	27	2	3	3	6	6	6	2	3	3	6	6	6	27	27	27	27	2	2	2	6	···	6	2	2	2	6	···	6

42 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	6	6
type	+	+	+				+	+	+	+	+	+					+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	S₃	S₃	D₆	D₆	D₆	C₃×S₃	C₃×S₃	S₃×C₆	S₃×C₆	He₃.₄S₃	C₂×He₃.₄S₃
kernel	C₂×He₃.₄S₃	He₃.₄S₃	C₂×C₉○He₃	C₂×C₉⋊S₃	C₉⋊S₃	C₃×C₁₈	C₃×C₁₈	C₂×He₃	C₂×3_-¹⁺²	C₃×C₉	He₃	3_-¹⁺²	C₁₈	C₃×C₆	C₉	C₃²	C₂	C₁
# reps	1	2	1	2	4	2	1	1	2	1	1	2	6	2	6	2	3	3

Matrix representation of C₂×He₃.₄S₃ ►in GL₆(𝔽₁₉)

18	0	0	0	0	0
0	18	0	0	0	0
0	0	18	0	0	0
0	0	0	18	0	0
0	0	0	0	18	0
0	0	0	0	0	18

0	0	18	1	0	0
1	1	17	18	0	0
0	0	18	0	1	0
0	0	18	0	0	1
0	0	18	0	0	0
1	0	18	0	0	0

0	18	0	0	0	0
1	18	0	0	0	0
1	0	18	18	0	0
0	18	1	0	0	0
1	0	0	0	18	18
0	18	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
1	1	18	18	0	0
1	1	0	0	18	18
0	0	0	0	1	0

14	17	0	0	0	0
2	12	0	0	0	0
2	0	12	17	0	0
0	17	2	14	0	0
2	0	0	0	12	17
0	17	0	0	2	14

0	18	0	0	0	0
18	0	0	0	0	0
0	0	0	0	18	0
18	18	0	0	1	1
0	0	18	0	0	0
18	18	1	1	0	0

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,1,0,0,0,1,0,1,0,0,0,0,18,17,18,18,18,18,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,1,0,1,0,18,18,0,18,0,18,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[1,0,0,1,1,0,0,1,0,1,1,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[14,2,2,0,2,0,17,12,0,17,0,17,0,0,12,2,0,0,0,0,17,14,0,0,0,0,0,0,12,2,0,0,0,0,17,14],[0,18,0,18,0,18,18,0,0,18,0,18,0,0,0,0,18,1,0,0,0,0,0,1,0,0,18,1,0,0,0,0,0,1,0,0] >;

C₂×He₃.₄S₃ in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3._4S_3

% in TeX

G:=Group("C2xHe3.4S3");

// GroupNames label

G:=SmallGroup(324,147);

// by ID

G=gap.SmallGroup(324,147);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,3171,453,2164,1096,7781]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=f^2=1,e^3=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,f*c*f=c^-1,d*e=e*d,d*f=f*d,f*e*f=c^-1*e^2>;

// generators/relations

G = C2×He3.4S3 order 324 = 22·34

Direct product of C2 and He3.4S3

G = C₂×He₃.₄S₃ order 324 = 2²·3⁴

Direct product of C₂ and He₃.₄S₃